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Sierpinski Gasket (Triangle) by Deterministic Recursion Algorithm

Below is an example of the output from a deterministic algorithm to generate Sierpinski Gasket also called Sierpinski Triangle.

Only certain width of the triangle will generate the picture (3, 5, 9, 17, 33, 65, 129...), for the example below width 129 was chosen.

This is not an example of a fractal, work is ongoing to create Sierpinski Deterministic Fractal, for this I need to switch to graphical output (non ASCII).

and generate Sierpinski Fractal via so called "Chaos Game", for an accessible explanation of how such a process works see ref. 1

The work later will continue to study IFS (Iterative Function System) using variety of random samples enhancing the algorithm with Variable Fractals, but need to start with

simplest possible model.

 

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Press any key to continue . . .

References

1. http://www.jcu.edu/math/vignettes/ChaosGame.htm

Entry #8

Superfractals - current progress on developing the framework

Began assembling the tools to model Superfractals within a given lottery game.

Motivation for the work is served from the ref 1. (especially section 6.3)

First Phase:

1. Development of  template program to gather lottery historical data.

Second Phase:

1. Model Superfractal IFS probability tables from the past lottery number/frequency transitions.

2. Perform IFS  iterations

3. Compare the outputs from IFS (Iterative Function System) to lottery historical data. (TBD. best fit function, distance measure etc)

4. Use Chaos Game and Monte Carlo methods to improve 3.

5. Look for emergence of Superfractal and/or ergodicity around the Superfractal.

Third Phase:

1. Project  (using fractal inerpolation) possible future iterations of lottery number/frequency transitions

Important Note: we are not trying to attempt to predict exact lottery numbers but looking for future lottery number's distributions in the number/frequency domain.

First Phase has been completed, source (superfractal.cpp), sample input data file and sample output is now available  (see ref. 2)

Sample output from the application (Phase 1):

Latest Draw     05/29/2009 23 30 36 39 48 +34 $35,000,000.00


White Ball Number/Frequency (Sorted by Number)

Number          01   02   03   04   05   06   07   08   09   10   11   12   13   14   15   16   17   18   19   20
Frequency      06   09   04   07   03   01   01   02   07   08   07   07   05   04   07   07   03   05   05   04

Number          21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40
Frequency      07   08   04   07   05   04   08   05   07   03   04   03   05   04   04   07   06   04   08   06

Number          41   42   43   44   45   46   47   48   49   50   51   52   53   54   55   56
Frequency      02   03   05   05   03   05   03   09   02   05   06   07   04   02   05   03

White Ball Number/Frequency (Sorted by Frequency)

Number          06   07   08   41   49   54   05   17   30   32   42   45   47   56   03   14   20   23   26   31
Frequency      01   01   02   02   02   02   03   03   03   03   03   03   03   03   04   04   04   04   04   04

Number          34   35   38   53   13   18   19   25   28   33   43   44   46   50   55   01   37   40   51   04
Frequency      04   04   04   04   05   05   05   05   05   05   05   05   05   05   05   06   06   06   06   07

Number          09   11   12   15   16   21   24   29   36   52   10   22   27   39   02   48
Frequency      07   07   07   07   07   07   07   07   07   07   08   08   08   08   09   09

Frequency      Count

01                  02
02                  04
03                  08
04                  10
05                  11
06                  04
07                  11
08                  04
09                  02

Press any key to continue . . .

References:

1. Superfractals - more detailed technical information - A FRACTAL VALUE RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY - MICHAEL BARNSLEY, JOHN HUTCHINSON, AND ÖRJAN STENFLO

http://www.superfractals.com/Superfractals/V-Variables.pdf

2. superfractal.cpp template application and supporting files. 

https://members.lotterypost.com/edge/index.htm

Entry #7

Superfractals - rapid computation of good approximations to random (and "fully" random) process

Reading this book currently (ref. 1) , eyeing application in the lottery (possibly) as the (ref. 3) points out:

"Areas of potential applications include computer graphics and rapid simulation of trajectories of stochastic processes The forward algorithm also enables rapid computation of good approximations to random (including “fully” random) processes, where previously there was no available efficient algorithm. "

In a similar venue of Monte Carlo methods, Superfractals promise to venture into areas where deterministic algorithms can not.

This book (ideas therein) is by far one of the most advanced on the topic. Fractals themselves emerge from set of deterministic rules (at least initially) but are supplemented by set of indeterministic cooefficients that set initial random trajectories and by the means of IFS (IFS - Iterated Function System) for the evolution to eventually emerge in some global symmetrical patterns.

Superfractals are new families of random fractals which are intermediate between the notions of deterministic and of standard random fractals and as such allow new type of sampling of the IFS evolution.

If it all sounds sophisticated, it is, ideas set forward in this book are already reverbarating cross multi-disciplinary sciences starting from bio-physics/ cosmology (ie. galaxy formations) / financial modeling / to number and game  theory.

It is my intention to eventually engage this theory (not just in lottery actually) to study random systems in general. 

References:

1. Superfractals - book by Michael Fielding Barnsley

http://www.amazon.com/SuperFractals-Michael-Fielding-Barnsley/dp/0521844932

2. Superfractals - website

http://www.superfractals.com/

3. Superfractals - more detailed technical information - A FRACTAL VALUE RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY - MICHAEL BARNSLEY, JOHN HUTCHINSON, AND ÖRJAN STENFLO

http://www.superfractals.com/Superfractals/V-Variables.pdf

Entry #6

Advanced Waves, Retrocausality and Consciousness - The Illusion of Time

In the 1945 paper by Wheeler and Feynman, Interaction with the Absorber as the Mechanism of Radiation (Rev. Mod. Phys. 17, 157 - 181 (1945)  for the first time aspects of charged accelerated particle and its EM waves of  a retarded type (propagating forward in time) and advanced (propagating backwards in time) type were studied.

Both waves are acceptable solutions to the classical Maxwell equations and there is nothing the relativistic Quantum Mechanics interpretations that forbids waves propagating backwards in time.

Recently a paper in the cognitive psychology field has been published on the subject of retrocausality (ref. 1), which is term a psychologist would use to define possibility of "advanced wave flowing backwards in time" concept in physics.

The normal theory of causality in the mind-world interaction makes use of forward flowing causality where one event precedes another, in the experiments quoted in the paper however, this causality is inefficient to explain results of the experiment. The result points to innate and strange ability of the mind to interact with the world where the concept of forward flow of time is challenged!

Quote from the paper:

"...

Retrocausality in REG (Random Event Generator) experiments. In 1979 the PEAR (Princeton Engineering Anomalies Research) laboratory was established under the direction of Robert Jahn, Dean of the University’s School of Engineering and Applied Sciences.

The purpose of this laboratory was to replicate and study the results obtained by a student which showed anomalous mind/machine interactions when using REG systems.

PEAR and a consortium of other universities have replicated these results. The anomalous mind/machine interaction which is observed is very simple: REG systems produce ultra-precise gaussian distributions, but when a subject tries to distort these distributions only by the expression of his intentionality, statistically significant deviations are observed.

Even more fascinating is the fact that those distributions which have been produced before the subjects’ expression of intentionality show an amplified effect.

The statistical significance of these “retrocausal” amplifications is p<0,000000001 (Jahn, 2005).

..."

References:

1. Advanced Waves, Retrocausality and Consciousness - paper by Antonella Vannini - Ph.D Student in Cognitive Psychology – University of Rome “La Sapienza”

http://www.hessdalen.org/sse/program/Antonella.pdf

2. Wheeler and Feynman, Interaction with the Absorber as the Mechanism of Radiation (Rev. Mod. Phys. 17, 157 - 181 (1945)

http://authors.library.caltech.edu/11095/1/WHErmp45.pdf

Entry #5

Padovan Sequence, Plastic Numbers and it's possible use in forecast models

Plastic Number derived from a unique real number solution to the equation x3 - x - 1 = 0 having the approximate value 1.324718 is intimately connected to the lower and upper limits of our normal ability
to perceive differences of size among three-dimensional objects. (see references 1 and 2)
Recently Plastic Number (or alternatively limit of the ratio of successive numbers in Padovan sequence) has found application in financial forecast model such as currency and commodity markets.

Its use is primarily in establishing price lower and upper limits as the function of past proportions occurring in currency/commodity price fluctuations. This effort is similar in how Fibonacci Ratios are used
in the financial trading.

Possible applications in the lottery forecast models could potentially exist providing that parallel analytical proportions do exists. This is very speculative and I could not find any references regarding
stochastic studies using Padovan Sequence. (except financial risk models mentioned before)

I speculate its potential success in commodity financial forecast stem from global proportions arising in the supply/demand cycles, such as natural climate paterns , human/animal migration patterns etc.

However  if the Padovan ratios demonstrate success in the currency model, this would rather pose an enigma, as the currency pricing can not be related (as far as i know) to any natural (world) proportions and cycles (supply and demand exists purely as abstraction of an underlying commodity/labour markets), in other words currency trading is very much like an un-bias stochastic game model (aka lottery), and it might be a hidden doorway to deeper view to the random...


Reference:

1. On Plastic Numbers in the Plane:

http://icgg2008.math.tu-dresden.de/abstracts/Spinadel-Redondo.pdf

1. Properties of Plastic Number:

http://www.daviddarling.info/encyclopedia/P/plastic_number.html

Entry #4

Distances between the winning numbers in Lottery. Combinatorics Study and Proof.

Recently a paper submitted to arxiv.org Combinatorics category came to my attention. (also under Annals of Pure and Applied Logic): "Distances between the winning numbers in Lottery" written by Konstantinos Drakakis and published in  16 March 2005

It provides a rigorous proof about Lottery (using Pick 6 game as a model): the winning 6 numbers (out of 49) in the game of the Lottery contain two consecutive numbers with a surprisingly high probability (almost 50%).

Almost 50%  still gives house an advantage, but is sufficiently large enough that it can not be ignored in generating prediction mumbers as in almost one game out of two the winning set of numbers contains two consecutive ones.

One simple strategy is to keep track of consecutive numbers in n-draws and calculate probability and predict accordingly according to the 50% rule.

Very recent draw in New Jersey Pick 5 game on April 1, 2009 with winning numbers: 11-12-13-25-33, prompted me to investigate consecutive numbers and look for some formal paper on the subject, the almost 50/50 chance of it occuring came as a big surprise to me...

References:

http://arxiv.org/PS_cache/math/pdf/0507/0507469v1.pdf           Distances between the winning numbers in Lottery by Konstantinos Drakakis

https://www.lotterypost.com/thread/189933/6                             Lotterypost thread of a draw in New Jersey Pick 5 game on April 1, 2009

Entry #3

Book Idea: Monte Carlo, Random Fields methods in Image Analysis

Title: Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction (Stochastic Modelling and Applied Probability) (Hardcover)

Author: Gerhard Winkler

Description:

This second edition of G. Winkler's successful book on random field approaches to image analysis, related Markov Chain Monte Carlo methods, and statistical inference with emphasis on Bayesian image analysis concentrates more on general principles and models and less on details of concrete applications.

Addressed to students and scientists from mathematics, statistics, physics, engineering, and computer science, it will serve as an introduction to the mathematical aspects rather than a survey. Basically no prior knowledge of mathematics or statistics is required.

The second edition is in many parts completely rewritten and improved, and most figures are new. The topics of exact sampling and global optimization of likelihood functions have been added. This second edition comes with a CD-ROM by F. Friedrich,containing a host of (live) illustrations for each chapter. In an interactive environment, readers can perform their own experiments to consolidate the subject.

Why?

Good ol' question why. What does image recognition have to do with lottery predictions you ask? first what does it have that is not in common!

For starters, image recognition is not about prediction, its about inference or classification of historical data.

A qualified image in nature, such an image of a tree can be categorized by finding repetitive features (such as leaves, trunk, tree apples etc) in lottery no such a repetitive features exists, no image recognition can be applied to lottery data as no repetitive matrix can be found (some will disagree on this point!)

However if an image is hard to categorize, it does not have distinctive features, image recognition becomes very hard (i.e computing some polygons by a deterministic algorithm and arriving with solutions to an equation in some n-space represented by some complex polynomial) and that's where stochastic methods such as Monte Carlo comes along, by a massive random sampling bombardment a target image is dissected and compared to some large database containing samples of vast number of possible shapes including complete shapes in a given classification (ie sequoia tree)

It can happen that image targeted for recognition fails even with Monte Carlo method. the failure (threshold of failure) is a function of incomplete sampling database and or function of computational time/space requirements needed to arrive with the solution (or approximation to the solution)

All images in theory should be recognizable providing that appropriate algorithm defining n-dimensional polygons exists or a an appropriate sampling exists in a database. The same is certain for a lottery game, a series of numbers can be in principle stored as some polynomial (coefficients ofwhich are ie winning lottery numbers) such a polynomial becomes an image/object in some n-dimensional space  (in a certain sense) or a database defined containing winning numbers (more common).

Can Monte Carlo be used in predictive sense?

Same way as image recognition methods can arrive with prediction about possible future state of an image (ie state of motion of a tree being perturbed by the wind) such a prediction would require ultra sophisticated algorithms but by Monte Carlo Stochastic methods motion of a tree can be extrapolated to the solutions that yet does not exist (future),  made possible by analyzing certain patterns that occured in the past, Monte Carlo can predict better than deterministic algorithm can especially in systems that are on the threshold of chaos.

Question than can Monte Carlo be used to predict lottery becomes a question of: is lottery a highly disordered system or is it a system that is in maximally disordred state (chaos) (in the case of latter Monte Carlo is of no value)

Reference:

http://www.amazon.com/Analysis-Random-Fields-Markov-Methods/dp/3540442138

Entry #2

Generating random arrangement of columns in sets of equal length

Based on the java package from Fermi Labs (randomX) that uses decay of Cæsium-137 particle as the source of random numbers,(hence quantum mechanical physical process for which no known algorithmic explanation exists - extending big thanks to JadeLottery for pointing this out to me!), created this small application to basically take any number of sets and length and create output consisting of sets of the same size but with random column permutations.

Program takes 4 inputs:

iterations   - number of times application should run complete full cycle
max           - length of set
sets            - number of sets
input          - 2 dimensional input array

and an one output: (displayed to the console):

output - 2 dimensional randomized output array

Example of an output generated by this application:

Input sets:

06-12-18-31-39-40-43-51-63-71
13-25-34-37-41-55-58-61-77-79
14-21-33-36-38-50-53-62-69-76

Output sets after 10 iterations:

13-12-18-37-41-50-53-51-63-76
06-21-34-31-38-55-58-61-77-71
14-25-33-36-39-40-43-62-69-79

The application could be used to study various strategies as to i.e. measuring how a non-random, pattern-based system predictions compare to predictions made by  a truly random process agent.

One could further extend it by assigning weight (bias) to the indvidual number(s) (lock) something I have not implemented keeping implementation to bare minimum.

Example provided will generate randomized pick-10 matrix-set after 10 iterations.

Hope this will be useful to anyone testing their prediction model(s)! 

source:

/* Generates random arrangement of columns in sets of equal length */


import randomX.*;
class randomXdemo {
public static void main(String[] args) {


int iterations = 10;
int max = 10;
int sets = 3;

int input[][] = {{6,12,18,31,39,40,43,51,63,71},
{13,25,34,37,41,55,58,61,77,79},
{14,21,33,36,38,50,53,62,69,76}};

/*
int max = 4;
int sets = 6;
int input[][] = {{1, 3, 2, 6},
{3, 9, 1, 5},
{4, 8, 9, 4},
{5, 6, 0, 1},
{6, 4, 8, 3},
{9, 1, 5, 2}};*/

/*
int max = 6;
int sets = 6;
int input[][] = {{3, 12, 14, 33, 49, 51},
{4, 6, 21, 25, 48, 50},
{5, 17, 20, 22, 29, 45},
{10, 18, 23, 42, 43, 55},
{15, 19, 27, 34, 36, 37},
{16, 30, 38, 44, 54, 56}};
*/

int output[][] = new int[sets][max];
display("Input sets:\n\n", input, sets, max);
randomX r = new randomLEcuyer();


for (int n = 0; n < iterations; n++) {
init(output, max, sets);
for (int i = 0; i < max; i++) {
for (int j = 0; j < sets; j++) {
output[find(r, 0, sets, i, output)][i] = input[j][i];
}
}
}
display("Output sets after " + iterations + " iterations:\n\n", output, sets, max);
}
static void init(int output[][], int max, int sets) {
for (int i = 0; i < max; i++) {
for (int j = 0; j < sets; j++)
output[j][i] = -1;
}
}
}
static int find(randomX r, int lBound, int uBound, int i, int output[][]) {
int n = 0;
do {
n = int2range(byte2int(r.nextByte()), lBound, uBound-1);
} while (output[n][i] != -1);
return n;
}
static int int2range(int i, int lBound, int uBound) {
return (int)(i % (uBound-lBound + 1)) + lBound;
}
static int byte2int(byte b) {
if (b < 0)
return (int)b + 0x100;
return b;
}
static void display(String title, int output[][], int sets, int max) {
int n = 0;
StringBuffer sb = new StringBuffer();
sb.append(title);
for (int j = 0; j < sets; j++) {
for (int i = 0; i < max; i++) {
n = output[j][i];
if (n < 10)
sb.append("0");
sb.append(n);
if (i < max-1)
sb.append("-");
}
sb.append("\n");
}
System.out.println(sb.toString());
}
}
/* source end */

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